Mathematical Sciences: Spectral and Fractal Geometry for Variational Elliptic Boundary Value Problems

流形的几何与流形上椭圆微分算子的不变量之间的关系是当代数学研究的主要统一主题之一。(例如，振动薄膜的固有频率构成了一种称为拉普拉斯算子的微分算符的频谱；知道了频谱，人们就可以对薄膜的形状做出很多推断。)具有分形边界的区域是如此不规则，以至于其维度不再是整数，这是最近将光谱信息与几何联系起来的一些突破的背景。特别是，由于拉皮德斯教授和他的一位法国同事的工作，现在人们知道可以从这样一个区域的拉普拉斯谱中恢复边界的尺寸。这项工作涉及到计算机实验和复杂的数学方法，在研究多孔介质和从分形表面散射波方面有应用。它的继续是该奖项支持的项目的主旨。更准确地说，拉皮德斯将研究具有各种边界条件的拉普拉斯算子本征值分布的渐近公式的锐化形式。目标是更清楚地说明边界的Minkowski维度在这种分布中的作用。关于伴随热半群的迹也有类似的公式。不仅研究区域的边界，而且研究区域本身是分形的情况。

The relationship between the geometry of a manifold and the invariants of the elliptic differential operators on it is one of the main unifying themes of contemporary mathematical research. (For instance, the natural frequencies of a vibrating membrane constitute the spectrum of a differential operator called the Laplacian; knowing the spectrum, one can infer a great deal about the shape of the membrane.) The case of a region with fractal boundary, so irregular that its dimension is no longer an integer, has been the setting for some recent breakthroughs relating spectral information to geometry. In particular, thanks to work of Professor Lapidus and a French colleague of his, it is now known that one can recover the dimension of the boundary from the spectrum of the Laplacian in such a region. This work, which has involved both computer experimentation and sophisticated mathematical technique, has applications to the study of porous media and the scattering of waves from fractal surfaces. Its continuation is the main thrust of the project supported by this award. More precisely, Lapidus will investigate sharpened forms of the asymptotic formula for the eigenvalue distribution of Laplace operators with various boundary conditions. The goal is to make yet clearer the role of the Minkowski dimension of the boundary in this distribution. There are analogous formulas for the trace of the associated heat semigroup. The case in which not only the boundary of the domain, but the domain itself, is fractal will be studied.